Topic: Application of Derivatives (Test 1)



Topic: Application of Derivatives
Q.1
The instantaneous rate of change at t = 1 for the function f (t) =  is
A. 2
B. -1
C. 0
D. 9
Answer : Option C
Explaination / Solution:

f(1)=e1+e1=0
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Q.2
The function f (x) =  – 2 x is increasing in the interval
A. x⩾1
B. x≠−1
C. x⩾−1
D. x≠1
Answer : Option A
Explaination / Solution:

 f(x) = x2- 2x
f'(x) = 2x - 2 = 2(x - 1) 
So , f( x) is increasing if 2(x-1) 0 , i.e.if x 1

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Q.3

The function f (x) =  strictly increases on


A. [0, 2]
B. (0,∞)
C. [ −−∞, 0]∪[2,∞)
D. none of these.
Answer : Option A
Explaination / Solution:

Here , 



i.e. if x(x – 2 )<0 i.e. if 0 < x < 2. Hence f is strictly increasing on [0,2]

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Q.4
Let g (x) be continuous in a neighbourhood of ‘a’ and g (a) ≠ 0. Let f be a function such that f ‘ (x) = g(x)  then
A. f is increasing at a if g (a) > 0
B. none of these
C. f is decreasing at a if g (a) >0
D. f is increasing at a if g (a) < 0
Answer : Option A
Explaination / Solution:

Since g is continuous at a , therefore , if g ( a ) > 0 , then there is a neighbourhood of a, say ( a-e , a+ e ) in which g ( x ) is positive .This means that f ‘ (x)>0 in this nhd of a and hence    f ( x ) is increasing at a.

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Q.5

The tangent to the parabola at the point  makes with the X – axis an angle of


A. 60
B. 30
C. 0
D. 45
Answer : Option D
Explaination / Solution:


Therefore , slope of tangent at ( 1 , ½ ) = 1. Hence , required angle is 
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Q.6
The equation of the tangent to the curve  at the point (0, 1) is
A. y + 1 = 2 x
B. y – 1 = 2 x
C. 1 – y = 2 x
D. none of these.
Answer : Option B
Explaination / Solution:



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Q.7

Tangents to the curve  at the points (1, 1) and ( – 1, 1)


A. parallel
B. intersecting but not at right angles
C. none of these
D. at right angles
Answer : Option D
Explaination / Solution:

 therefore , slope of tangent at (1,1) = - 1 and the slope of tangent at ( - 1 ,1 )= 1 .

Now product of the slopes=1.-1= -1

Hence , the two tangents  are at right angles.


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Q.8
The point on the curve  where tangent makes an angle of  with the X – axis is
A. (14,12)
B. (12,14)
C. (2, – 2)
D. (4, 2)
Answer : Option A
Explaination / Solution:



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Q.9
Let f (x) =  then f (x) is
A. an even function
B. a decreasing function.
C. an increasing function
D. an odd function
Answer : Option C
Explaination / Solution:




Hence an increasing function.


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Q.10
Let f (x) =  – 4x, then
A. f is increasing in [−∞,1)
B. f is increasing in [1,∞)
C. f is decreasing in [1,∞)
D. none of these.
Answer : Option B
Explaination / Solution:

f (x) = x4 – 4x

f'(x) = 4(x3) - 4 = 4(x 3- 1) = 4 {(x2) + x + 1)} (x-1) 
= f ‘(x) > 0,if,(x - 1) > 0,
and f ‘(x) <0,  if (x -1 ) < 0.
So,f is decreasing on ( - ,1]

and f is increasing on [1,)


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